Sharp inequality proven for symmetric functions on a 4D sphere.
problem Proving a sharp Beckner's inequality for axially symmetric functions on S4. method Utilized pointwise properties of Gegenbauer polynomials.
result Sharp Beckner's inequality established for axially symmetric functions on S4. The article improves Beckner's inequality for axially symmetric functions on the n-dimensional sphere.
problem Improving Beckner's inequality for axially symmetric functions on Sn. method Uniqueness and existence results for Q-curvature type equations with a Paneitz operator on Sn for axially symmetric functions. result Improved Beckner's inequality for axially symmetric functions on Sn. In this paper, a new approach of defining Steiner symmetrization of coercive convex functions is proposed and some fundamental properties of the new Steiner symmetrization are proved. Further, using the new Steiner symmetrization, we give a different approach to prove a functional version of the Blaschke-Santalo inequa…
The Stanley chromatic symmetric function XG of a graph G is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology of graded Sn-modules, whose graded Frobenius series FrobG(q,t) reduces to …
Researchers create explicit p-harmonic functions on specific symmetric spaces.
problem Constructing explicit p-harmonic functions on compact Riemannian symmetric spaces.
method Explicit construction of complex-valued p-harmonic functions on specific symmetric spaces and their duals.
result Explicit p-harmonic functions constructed on SU(n)/SO(n), Sp(n)/U(n), SO(2n)/U(n), SU(2n)/Sp(n) and their duals.
We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.
We consider the problem of learning regression functions from pairwise data when there exists prior knowledge that the relation to be learned is symmetric or anti-symmetric. Such prior knowledge is commonly enforced by symmetrizing or anti-symmetrizing pairwise kernel functions. Through spectral analysis, we show that …
Harmonic functions on compact symmetric spaces exhibit strong convexity properties.
problem Understanding the convexity of harmonic functions on compact symmetric spaces.
method Analyzing the nonnegativity of the Laplacian powers of harmonic functions.
result Harmonic functions on compact symmetric spaces have nonnegative Laplacian powers, demonstrating strong convexity.
Symmetrizes loss functions to improve neural network robustness against noisy labels.
problem Designing robust loss functions for noisy labels in neural networks.
method Symmetrization of multi-class loss functions, focusing on cross-entropy and unhinged loss.
result The multi-class unhinged loss is the unique convex symmetric loss under suitable assumptions.
The paper finds determinantal expressions for certain symmetric space integrals.
problem Finding compact expressions for integrals on symmetric spaces.
method Expressing integrals as determinants or Pfaffians for K-invariant functions. result Determinantal expressions for specific symmetric space integrals.
New neural network models learn symmetric functions of varying input sizes.
problem Learning symmetric functions with varying input sizes.
method Functional perspective on neural networks, treating symmetric functions as functions over probability measures.
result Established approximation and generalization bounds for shallow architectures that extend across input sizes.
New neural networks respect symmetries in symmetric tensors, improving efficiency and generalization.
problem Learning from symmetric tensors efficiently and respecting their inherent symmetries.
method Developed two characterizations of linear permutation equivariant functions between symmetric power spaces of R^n.
result These functions are highly data efficient compared to standard MLPs and generalize well to different sizes of symmetric tensors.
The aim of this paper is to study the Lelong number, the integrability index and the Monge-Ampère mass at the origin of an S1-invariant plurisubharmonic function on a balanced domain in Cn under the Schwarz symmetrization. We prove that n times the integrability index is exactly the Lelong number of th…
Study finds lower bounds for energy on fibred manifolds using fiberwise symmetrization.
problem Finding lower bounds for energy functionals on fibred manifolds.
method Established a framework for fiberwise symmetrization to find lower bounds.
result Proved a comparison theorem for the first eigenvalue of the Laplacian on warped product manifolds.
Paper refines Talagrand inequality on Euclidean spaces.
problem Improving Talagrand inequality for Euclidean spaces.
method Symmetrization and alternative proof methods.
result Several refined functional inequalities derived.
We study the Selberg zeta and the theta function associated to bundles over even-dimensional locally symmetric spaces of rank one.
Extends functions on symmetric spaces to analytic functions.
problem Extending functions on symmetric spaces to analytic functions.
method Harmonic analysis on symmetric spaces and representation theory of groups.
result Proves Whitney type extension theorems for symmetric spaces.
Paper proves March's criterion for transience on symmetric manifolds.
problem Determining transience on rotationally symmetric manifolds.
method Analyzes bounded non-constant harmonic functions and Dirichlet problem at infinity.
result March's criterion is necessary and sufficient for transience.
Symmetric critical points lead to symmetry breaking in neural networks.
problem Understanding symmetry in critical points of invariant functions.
method Analyzing the symmetry of critical points and their neighbors in invariant nonconvex functions.
result Symmetric critical points in invariant nonconvex functions are generically followed by symmetry breaking adjacent points.
Improved Beckner's inequality for axially symmetric functions on S^4.
problem Proving axially symmetric solutions to a constant Q-curvature type equation must be constant.
method Analyzing constant Q-curvature type equations on S^4, using Pohozaev-type identities and bifurcation methods.
result Improved Beckner's inequality for axially symmetric functions on S^4.
Constructs explicit harmonic functions and morphisms on complex and quaternionic Grassmannians.
problem Creating explicit solutions for p-harmonic functions and harmonic morphisms. method Using joint eigenfunctions of the Laplace-Beltrami and conformality operators.
result Induces solutions on dual non-compact Riemannian symmetric spaces.
In this note, we discuss symmetric brackets on skew-symmetric algebroids associated with a metric structure. Given a pseudo-Riemannian metric structure, we describe symmetric brackets induced by connections with totally skew-symmetric torsion in the language of Lie derivatives and differentials of functions. In particu…
This paper provides an explicit form for symmetric differentials and their corresponding holomorphic functions.
problem Understanding the correspondence between symmetric differentials and L2 holomorphic functions on quotient spaces. method Explicit description of the correspondence between symmetric differentials and weighted L2-holomorphic functions. result Derivation of several applications based on the explicit form of the correspondence.
Harmonic morphisms and p-harmonic functions constructed on symmetric spaces.
problem Constructing harmonic morphisms and p-harmonic functions on symmetric spaces.
method Using Cartan embedding and related maps to relate tension field and conformality operator.
result Simple formulae relating tension field and conformality operator on symmetric spaces to those on their images.
Zeta functions for non-unitary twists are shown to have analytic continuation.
problem Analytic continuation of zeta functions for non-unitary twists.
method Analytic continuation for compact locally-symmetric spaces with non-unitary twists.
result Zeta functions admit analytic continuation as meromorphic functions.
Upper bound found for divergence-free Killing 2-tensors on manifolds.
problem Bounding the space of divergence-free symmetric Killing 2-tensors.
method Witten deformation and Morse function analysis.
result Explicit calculation of dimension for p=2. Extends Milnor's criterion to biharmonic functions.
problem Deciding surface type for biharmonic functions.
method Generalizes Milnor's criterion to biharmonic functions.
result Characterizes whether a surface is hyperbolic or parabolic for biharmonic functions.
This work improves understanding of symmetrizing Bregman divergences on positive definite matrices.
problem Understanding which mean to use for symmetrizing Bregman divergences on positive definite matrices.
method Axiomatic definition of mean functionals and variational principles over the cone of positive definite matrices.
result The arithmetic mean is canonical for forward symmetrization, and the arithmetic, log-Euclidean, and harmonic means for reverse symmetrization.
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
Transformers tend to learn more symmetric functions in sequence data.
problem Understanding inductive bias in Transformers with infinitely over-parameterized models.
method Analyzing Transformers in the Gaussian process limit, using representation theory of the symmetric group.
result Transformers are biased towards more permutation symmetric functions, and this can be quantitatively predicted.
Complex valued analytic torsion and dynamical zeta function studied on locally symmetric spaces.
problem Analyzing the Ruelle dynamical zeta function on locally symmetric spaces with flat vector bundles.
method Meromorphic extension and regularisation of the dynamical zeta function, relating it to the complex valued analytic torsion.
result The leading term of the dynamical zeta function at zero is related to the regularised determinant of the flat Laplacian.
We consider the generalized Segal-Bargmann transform, defined in terms of the heat operator, for a noncompact symmetric space of the complex type. For radial functions, we show that the Segal-Bargmann transform is a unitary map onto a certain L^2 space of meromorphic functions. For general functions, we give an inversi…
Study on symmetric domains with Bergman metric properties.
problem Properties of Bergman metrics in symmetric domains.
method Combining Hermitian symmetric spaces theory, Kähler immersions, and analytic/pluripotential tools.
result Rigidity results for Bergman metrics in bounded domains.
The action of origin-preserving diffeomorphisms on a space of jets of symmetric connections is considered. Dimensions of moduli spaces of generic connections are calculated. Poincaré series of the geometric structure of symmetric connection is constructed, and shown to be a rational function.
Deep networks can memorize random labels; symmetric loss improves this.
problem Deep networks can memorize random labels, ignoring standard regularization.
method Empirical studies with MNIST and CIFAR-10 datasets, formal definition of robustness.
result Symmetric loss function improves network's ability to resist memorization.
Symmetric function LM,N lifts torus link homology.
problem Computing the triply-graded Khovanov-Rozansky homology of torus links.
method Defined a symmetric function LM,N and showed it satisfies a recursion for torus link homology. result Triply-graded Khovanov-Rozansky homology of torus links is a specialization of LM,N. The nonzero level sets in n-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the nth power of the function. The exponentials of the characteristic polynomials of certa…
The sinh-Gordon equation is solved on finite, symmetric graphs.
problem Solving the sinh-Gordon equation with nonzero prescribed functions on finite graphs.
method Uniform a priori estimate to define topological degree, case-by-case calculation of degree, classical sinh-Gordon equation analysis.
result The classical sinh-Gordon equation with nonzero prescribed function is always solvable on finite, symmetric graphs.
Study on stability of Einstein metrics on symmetric spaces.
problem Stability of Einstein-Hilbert functional on compact symmetric spaces.
method Classification of irreducible representations and use of Casimir eigenvalues.
result Proves stability of Einstein metrics on quaternionic and Cayley projective plane, instability on other quaternionic Grassmannians.
The oriented framed Homfly skein C of the annulus provides the natural parameter space for the Homfly satellite invariants of a knot. It contains a submodule C+ isomorphic to the algebra of the symmetric functions. We collect and expand formulae relating elements expressed in terms of symmetric functions to Turaev's ge…
We investigate existence and stability of rotationally symmetric critical immersions of variational problems of higher order which were considered by Nitsche.
A new approach to Morse theory using folded ribbon trees.
problem Applying Morse theory on symmetric products of surfaces.
method Introducing an A-infinity category with objects as κ-tuples of Morse functions, and showing conditions for the endomorphism to be a Hecke algebra.
result The endomorphism of a specific type of κ-tuple of Morse functions on T*R^2 is the Hecke algebra associated to the symmetric group.
Paper computes link determinants using Fourier-Hadamard transforms.
problem Computing determinants of complex link structures.
method Fourier-Hadamard transforms of Boolean functions.
result Determinant of centrally symmetric links with even components equals zero.
We prove that a function on an irreducible compact symmetric space M, which is not a sphere, is determined by its integrals over the shortest closed geodesics in M. We also prove a support theorem for the Funk transform on rank one symmetric spaces which are not spheres.
Paper introduces symmetric divergence link models for probability distributions.
problem Symmetric divergence measures for probability distributions.
method Two general classes of link models: one for survival functions and another for cumulative probability distribution functions.
result Advantages of symmetric divergence measures over asymmetric measures for model averaging and feature assessment.
This contribution deals with the generalized symmetric FastICA algorithm in the domain of Independent Component Analysis (ICA). The generalized symmetric version of FastICA has been shown to have the potential to achieve the Cramér-Rao Bound (CRB) by allowing the usage of different nonlinearity functions in its paralle…
Locally classifies 4D spherical symmetric Finsler spaces.
problem Classifying 4D spherical symmetric Finsler spaces.
method Local classification of pseudo-Finsler Berwald structures.
result Six classes of non-Riemannian SO(3)-symmetric pseudo-Finsler Berwald functions.
The paper classifies spherically symmetric Finsler metrics as Berwaldian or Riemannian.
problem Classifying spherically symmetric Finsler metrics.
method Proving all spherically symmetric Landsberg surfaces are Berwaldian and modifying the classification of spherically symmetric Finsler metrics.
result All Berwald spherically symmetric metrics of dimension n≥3 are Riemannian or given by a specific formula.